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17.12 The notion of test curves908139B245F5

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Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\theta $ be a smooth closed real $(1,1)$-form on $X$ representing a big cohomology class.

Recall that the notion of model potentials is defined in definition 16.3.18.

Definition 17.12.1.C8399F09D86D A test curve $\Gamma $ in $\mathrm{PSH}(X,\theta )$ consists of a real number $\Gamma _{\max }$ together with a map $(-\infty ,\Gamma _{\max })\rightarrow \mathrm{PSH}(X,\theta )$ denoted by $\tau \mapsto \Gamma _{\tau }$ satisfying the following conditions:

  1. D665BBFA648A

    The map $\tau \mapsto \Gamma _{\tau }$ is concave and decreasing;

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    $P_{\theta }[\Gamma _{\tau }]=\Gamma _{\tau }$;

  3. 28110BFBC4C9

    the potential

    E3D67CAF2F52 17.12.2
    \begin{equation} \Gamma _{-\infty }\coloneqq \operatorname *{sup}_{\tau <\Gamma _{\max }}{}\! {}^{*}\Gamma _{\tau } \end{equation}

    satisfies

    \[ \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{-\infty }\right)^ n>0. \]

Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential. The set of test curves $\Gamma $ with $\Gamma _{-\infty }=\phi $ is denoted by $\mathrm{TC}(X,\theta ;\phi )$.

The union of all $\mathrm{TC}(X,\theta ;\phi )$'s for various model potentials $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ is denoted by $\mathrm{TC}(X,\theta )_{>0}$.

By 2, $\sup _{X}\Gamma _{\tau }=0$ for each $\tau <\Gamma _{\max }$. So $\Gamma _{-\infty }\in \mathrm{PSH}(X,\theta )$ by proposition 5.14.2.

Thanks to corollary 13.13.12 and 3, there is $C<\Gamma _{\max }$ so that whenever $\tau <C$, we have $\Gamma _{\tau }\in \mathrm{PSH}(X,\theta )_{>0}$. Then it follows that $\Gamma _{-\infty }$ is a model potential by proposition 16.4.9.

Remark 17.12.3.3D73AB2E32CB Sometimes it is convenient to extend $\Gamma _{\tau }$ to $\tau \geq \Gamma _{\max }$ as well. This can be done as follows: for $\tau >\Gamma _{\max }$, we set $\Gamma _{\tau }\equiv -\infty $. For $\tau =\Gamma _{\max }$, we set

\[ \Gamma _{\tau }\coloneqq \inf _{\tau '<\Gamma _{\max }}\Gamma _{\tau '}\in \mathrm{PSH}(X,\theta ). \]

Note that $\Gamma _{\tau }\in \mathrm{PSH}(X,\theta )$ by corollary 5.17.5.

In general, for any $x\in X$, the map $\tau \mapsto \Gamma _{\tau }(x)$ falls into one of the following three categories:

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    $\Gamma _{\tau }(x)\equiv -\infty $;

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    $\Gamma _{\tau }(x)>-\infty $ for all $\tau \leq \Gamma _{\max }$;

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    there is $\tau (x)\in (-\infty ,\Gamma _{\max }]$ so that $\Gamma _{\tau }(x)>-\infty $ for all $\tau \in (-\infty ,\tau (x))$, and $\Gamma _{\tau }(x)=-\infty $ for all $\tau \geq \tau (x)$.

The last two possibilities are illustrated in 17.12.4.

Mathematical diagram from the source
9BE4C00F2E1517.12.4 The last two pointwise possibilities for $\tau \mapsto \Gamma _\tau (x)$.

Note that 2 is the generic situation. It occurs whenever $x\notin \{ \Gamma _{\Gamma _{\max }}=-\infty \} $. In particular, when this situation occurs, $(-\infty ,\Gamma _{\max })\ni \tau \mapsto \Gamma _{\tau }(x)$ is finite concave, and hence continuous. As a consequence, for any $\tau \in (-\infty ,\Gamma _{\max })$, we have

5EFADBE6069E 17.12.5
\begin{equation} \Gamma _{\tau }=\inf _{\tau '<\tau }\Gamma _{\tau '},\quad \Gamma _{\tau }=\operatorname *{sup}_{\tau '\in (\tau ,\Gamma _{\max })}{}\! {}^{*}\Gamma _{\tau '} \end{equation}

almost everywhere and hence everywhere due to proposition 5.14.6.

Recall that according to our general principle, we speak of model potentials only when a potential has positive mass.

Lemma 17.12.6.348ACDBCB24E Assume that $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$. Then for each $\tau <\Gamma _{\max }$, we have

F4B7B1DBBE89 17.12.7
\begin{equation} \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{\tau }\right)^ n>0. \end{equation}

In particular, $\Gamma _{\tau }$ is a model potential in $\mathrm{PSH}(X,\theta )_{>0}$.

The notations in the proof below are summarized in 17.12.8.
Mathematical diagram from the source
20FE43CB91F917.12.8 The test curve $\Gamma $.

Proof. Fix $\tau \in (-\infty ,\Gamma _{\max })$.

By assumption, $\Gamma _{-\infty }$ has positive mass. By corollary 13.13.12, we have

\[ \int _ X \theta _{\Gamma _{-\infty }}^ n=\lim _{\tau \to -\infty }\int _ X \theta _{\Gamma _{\tau }}^ n. \]

In particular, for a sufficiently small $\tau _0<\tau $, we have

\[ \int _ X \theta _{\Gamma _{\tau _0}}^ n>0. \]

Now take $\tau '\in (\tau , \Gamma _{\max })$ and $t\in (0,1)$ so that

\[ \tau =(1-t)\tau '+t\tau _0. \]

From the concavity of $\Gamma $, we find that

\[ \Gamma _{\tau }\geq (1-t)\Gamma _{\tau '}+t\Gamma _{\tau _0}. \]

By the monotonicity theorem theorem 13.13.11 and the multi-linearity of the non-pluripolar products proposition 13.11.4 6,

\[ \int _ X \theta _{\Gamma _{\tau }}^ n\geq \int _ X \theta _{(1-t)\Gamma _{\tau '}+t\Gamma _{\tau _0}}^ n\geq t^ n \int _ X \theta _{\Gamma _{\tau _0}}^ n>0 \]

and (17.12.7) follows. $\square$

Proposition 17.12.9.B7CD1B90CFDA Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$. Then the map

\[ [-\infty ,\Gamma _{\max })\rightarrow \mathbb {R},\quad \tau \mapsto \log \int _ X \theta _{\Gamma _{\tau }}^ n \]

is continuous and concave on $(-\infty ,\Gamma _{\max })$.

Proof. The concavity of this function follows from theorem 13.13.5 and theorem 13.13.11. The continuity at $-\infty $ is a consequence of corollary 13.13.12. The continuity on $(-\infty ,\Gamma _{\max })$ follows from the fact that finite concave functions on open intervals are continuous. $\square$

Definition 17.12.10.69529F41AA18 Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential.

A test curve $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$ is said to be bounded if for $\tau $ small enough, $\Gamma _{\tau }=\phi $. The subset of bounded test curves in $\mathrm{TC}(X,\theta ;\phi )$ is denoted by $\mathrm{TC}^{\infty }(X,\theta ;\phi )$. In this case, we write

3578A2265BDC 17.12.11
\begin{equation} \Gamma _{\min }\coloneqq \sup \{ \tau \in \mathbb {R}:\Gamma _{\tau }=\phi \} . \end{equation}

A test curve $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$ is said to have finite energy if

942AC971E484 17.12.12
\begin{equation} \mathbf{E}^{\phi }(\Gamma )\coloneqq \Gamma _{\max } \int _ X \theta _{\phi }^ n+\int _{-\infty }^{\Gamma _{\max }} \left(\int _ X\theta _{\Gamma _{\tau }}^ n-\int _ X \theta _{\phi }^ n\right)\, \mathrm{d}\tau >-\infty . \end{equation}

When $\phi =V_{\theta }$, we write $\mathbf{E}$ instead of $\mathbf{E}^{\phi }$.

The subset of test curves with finite energy in $\mathrm{TC}(X,\theta ;\phi )$ is denoted by $\mathrm{TC}^1(X,\theta ;\phi )$.

Example 17.12.13.1A17E2EECB2D Given $\varphi \in \mathrm{PSH}(X,\theta )$, there is a canonically associated test curve $\Gamma ^{\varphi }\in \mathrm{TC}^{\infty }(X,\theta ;V_{\theta })$: Set $\Gamma ^{\varphi }_{\max }=0$ and

\[ \Gamma ^{\varphi }_{\tau }= \begin{cases} V_{\theta },& \textup{if }\tau \leq -1;\\ P_{\theta }\left[ (1+\tau )\varphi -\tau V_{\theta } \right],& \textup{if }-1<\tau <0. \end{cases} \]

Note that $\Gamma ^{\varphi }$ is indeed a test curve, as follows from proposition 16.4.2. Indeed, the concavity and monotonicity in $\tau $ follow from proposition 16.4.2, and $\Gamma ^\varphi _{-\infty }=V_\theta $ has positive mass.

We first observe that the notion of test curves does not really depend on the choice of $\theta $ within its cohomology class.

Proposition 17.12.14.B015D2EA4B4F Let $\theta '$ be another smooth closed real $(1,1)$-form on $X$ representing the same cohomology class as $\theta $. Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential. Let $\phi '\in \mathrm{PSH}(X,\theta ')_{>0}$ be the unique model potential satisfying $\phi \sim \phi '$.

Then there is a canonical bijection

\[ \mathrm{TC}(X,\theta ;\phi )\xrightarrow {\sim } \mathrm{TC}(X,\theta ';\phi '). \]

This bijection induces the following bijections:

\[ \mathrm{TC}^1(X,\theta ;\phi )\xrightarrow {\sim }\mathrm{TC}^1(X,\theta ';\phi '),\quad \mathrm{TC}^{\infty }(X,\theta ;\phi )\xrightarrow {\sim }\mathrm{TC}^{\infty }(X,\theta ';\phi '). \]

These bijections satisfy the obvious cocycle conditions.

Proof. Choose $g\in C^{\infty }(X)$ such that $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}g$. Given any $\Gamma \in \mathrm{TC}(X,\theta ;\phi )$, we observe that $\Gamma '\colon (-\infty ,\Gamma _{\max })\rightarrow \mathrm{PSH}(X,\theta ')$ defined as

\[ \tau \mapsto P_{\theta '}[\Gamma _{\tau }-g] \]

lies in $\mathrm{TC}(X,\theta ';\phi ')$. In fact, the concavity and monotonicity of $\Gamma '$ follow from proposition 16.4.2. Moreover, applying proposition 16.4.9 to the increasing net obtained as $\tau \to -\infty $ gives

\[ \Gamma '_{-\infty }=P_{\theta '}[\Gamma _{-\infty }-g]=P_{\theta '}[\phi -g]=\phi '. \]

Thus $\Gamma '\in \mathrm{TC}(X,\theta ';\phi ')$.

Moreover, the choice of $g$ is irrelevant since for any other choice of $g$, say $g'$, we have

\[ \Gamma _{\tau }-g\sim \Gamma _{\tau }-g' \]

for all $\tau <\Gamma _{\max }$. The inverse construction is obtained by replacing $g$ with $-g$. The boundedness condition is clearly preserved. Finally, by proposition 16.3.10, for every $\tau <\Gamma _{\max }$,

\[ \int _ X(\theta '+\mathrm{dd}^{\mathrm{c}}\Gamma '_\tau )^ n=\int _ X(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _\tau )^ n, \]

and the same holds for $\phi $ and $\phi '$. Hence the finite energy condition is preserved. The cocycle condition is immediate from the uniqueness of the model representative in a fixed singularity type. $\square$

Proposition 17.12.15.2059587BD91B Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$. Then the pointwise pull-back induces a bijection

\[ \pi ^*\colon \mathrm{TC}(X,\theta ;\phi )\xlongrightarrow {\sim } \mathrm{TC}(Y,\pi ^*\theta ;\pi ^*\phi ). \]

Proof. The bijectivity on the underlying potentials follows from proposition 5.17.8. The compatibility with model potentials follows from proposition 16.4.1. These two facts also preserve concavity, monotonicity and the value of $\Gamma _{-\infty }$, hence give the asserted bijection of test curves. $\square$

Next we verify the closedness of a test curve as a family of concave functions, so that no pathology arises in the Legendre transforms considered below. The notion of closedness is recalled in definition 21.11.15.

Proposition 17.12.16.7AF1F81E928C Let $\Gamma $ be a test curve in $\mathrm{PSH}(X,\theta )$. For each $x\in X$, the map $\mathbb {R} \ni \tau \mapsto \Gamma _{\tau }(x)$ is a closed concave function. Moreover, the map is proper as long as $\Gamma _{\Gamma _{\max }}(x)\neq -\infty $.

Proof. We prove closedness. Fix $x\in X$. Assume that $\Gamma _{\tau }(x)\neq -\infty $ for some $\tau \in \mathbb {R}$. We only need to argue the upper semicontinuity of $\tau \mapsto \Gamma _{\tau }(x)$. The upper semicontinuity is clear at $\tau \geq \Gamma _{\max }$, so we are reduced to prove the following:

E703D3F2D79D 17.12.17
\begin{equation} \Gamma _{\tau }=\inf _{\tau '<\tau }\Gamma _{\tau '} \end{equation}

for any $\tau <\Gamma _{\max }$. Take $\tau ''\in (\tau ,\Gamma _{\max })$. Outside the polar locus of $\Gamma _{\tau ''}$, we know that (17.12.17) holds by continuity of real-valued concave functions. So (17.12.17) holds everywhere by proposition 5.14.6.

The final assertion follows from properness of a finite closed concave function on its effective domain. $\square$

Definition 17.12.18.37432F25600E Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\omega $ be a smooth closed real positive $(1,1)$-form. Then we define $P_{\theta +\omega }[\Gamma ]\in \mathrm{TC}(X,\theta +\omega )_{>0}$ as follows:

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    Define

    \[ P_{\theta +\omega }[\Gamma ]_{\max }=\Gamma _{\max }; \]
  2. 240BD675E3E0

    for each $\tau <\Gamma _{\max }$, define

    \[ P_{\theta +\omega }[\Gamma ]_{\tau }=P_{\theta +\omega }[\Gamma _{\tau }]. \]
It follows from proposition 16.4.2 that the family $P_{\theta +\omega }[\Gamma ]$ is concave and decreasing. Moreover, since $\omega $ is positive,
\[ \int _ X(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma _\tau )^ n \geq \int _ X(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _\tau )^ n>0 \]

for every $\tau <\Gamma _{\max }$, and hence $P_{\theta +\omega }[\Gamma ]_\tau $ is a model potential by corollary 16.3.20. Thus $P_{\theta +\omega }[\Gamma ]\in \mathrm{TC}(X,\theta +\omega )_{>0}$.

Proposition 17.12.19.724BF0CAE002 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\omega $ be a closed real smooth positive $(1,1)$-form on $X$. Then

\[ P_{\theta +\omega }[\Gamma ]_{-\infty }=P_{\theta +\omega }[\Gamma _{-\infty }]. \]

Proof. This follows from proposition 16.4.9. $\square$

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